Application of the Bonus-Malus System in Microcredit

Loan Repayment Problems in a Microﬁnance Institution
Adrian R. Llamado
BSAM
Roel D. Valenzuela
BSAM
October 14, 2013
Abstract
This study will primarily focus on the application of a bonus-malus system (BMS), which is more
commonly used in insurance policies, in minimizing loan repayment problems of a microﬁnance
institution. It will attempt to analyze the scheme of repayment process in a microﬁnance ﬁrm X and
determine factors that contribute to the microcredit loan repayment problems. From these factors,
new loan policies consisting of bonus-malus payments will then be formulated and introduced as
an improvement to the previous system that would minimize loan repayment problems.
1 Technical Description
1.1 Background of the Study
Microcredit is the lending side of microﬁnance. Basically, it is a system in which an individ-
ual can borrow money as a starting capital to start a business with little to no collateral.
Becoming increasingly popular in third world countries, microﬁnance institutions (MFI’s)
have been reported able to improve the economic and social status of the poor since it pro-
vides a working opportunity for them (Mokhtar et al., 2012). Microcredit has also been
praised by many as a tool for women empowerment, particularly in India, although some
have diﬀerent arguments on the matter (Mayoux, 2000). Issues of over-indebtedness of bor-
rowers could not be fully avoided, and thus pose a threat for the microﬁnance instituton to
have serious losses. This results to some institutions resorting to oppressive tactics to secure
the repayment from their borrowers.
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The bonus-malus system (BMS) is a stochastic process, an application of Markov chains
more prominently used in insurance policies. Within the context of loans, the system can be
used as such: for example, the lender (the microﬁnance institution), either gives the borrower
a bonus (i.e. a decrease in loan rate) or a malus (i.e. an increase in loan rate) based on the
borrower’s loan repayment history.
1.2 Statement of the Problem
The primary goal of the study is to determine factors contributing to the loan repayment
problems in a speciﬁc microﬁnance ﬁrm X and to develop optimal bonus-malus repayment
policies that would minimize these loan repayment problems.
1.3 Objectives
Speciﬁcally, the study aims to:
1. Construct a mathematical model to analyze the loan repayment scheme of microﬁnance
institution X.
2. Identify the signiﬁcant factors aﬀecting the loan repayment problems of borrowers of
microﬁnance ﬁrm X using statistical methods.
3. Formulate a unique bonus-malus model of repayments that would be ﬁtting for a
speciﬁc borrower’s classiﬁcation in terms of the indetiﬁed factors.
4. Evaluate the quality of the formulated models in terms of the measure of adequacy of
bonus-malus systems.
5. Develop new loan repayment policies based on the formulated models.
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1.4 Signiﬁcance of the Study
If this study proves to be successful, and if the microﬁnance institution X adopts the formu-
lated policies in the study, their risk of bankruptcy in the long run might be alleviated while
its borrowers’ risk of over-indebtedness will be reduced. The results of the study can also
prompt other lending institutions to develop similar policies to improve their transactions,
thus beneﬁtting the economy as a whole. Furthermore, the study will be the starting point
to open new possibilities where the bonus-malus system can be applied aside from insurance.
1.5 Review of Related Literature
1.5.1 “Microcredit models and Yunus equation” (2010) by L´eo Aug´e, Aurore
Lebrun, and Ana¨ıs Piozin
In this project, Aug´e et al. studied an example of a Yunus model of microcredit loan, given
by the equation:
1000 =
50

i=1
22e
−rtn
(1)
where 1000 is the amount borrowed, 22 is the amount to be repayed in one week (both
amounts are in Bangladesh Taka) for 50 weeks, r is the continuous rate of interest, and
t
n
=
(X
1
+X
2
+...+X
5
0)
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and X
n
is the reimbursement time for the n
th
week repayment to be
made.
Aug´e et al. assumed that X
n
follows a geometric distribution and simulated the interest
rates of the distribution. They then applied the same process with the necessary reﬁnements
to a particular loan repayment scheme of the microﬁnance company Cetelem in Cambodia
and the results showed that the distributions of the interest rate between people with loan
repayment problems and those without loan repayment problems are spread out signiﬁcantly,
and that those who pay very late are penalized with relatively high rates of interest, causing
them to suﬀer more from overindebtedness.
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1.5.2 “Determinants of microcredit loans repayment problem among microﬁ-
nance borrowers in Malaysia” (2012) by Suraya Hanim Mokhtar, Gilbert
Nartea, and Christopher Gan
In this study, Mokhtar et al. investigated the factors inﬂuencing the loan repayment problems
in TEKUN and YUM microﬁnance institutions in Malaysia. They surveyed borrowers from
these two ﬁrms and using logistic regression, found out that the determinants of the loan
repayment problems are the age, gender, and business-type of the borrower, as well as the
mode of repayment and repayment amount of the institution.
1.5.3 “Marketing and Bonus Malus Systems” (2003) by Sandra Pitrebois, Michel
Denuit, and Jean-Francois Walhin
In this article, Pitrebois et al. obtained data on the claims frequency of policyholders in an
insurance company with respect to factors such as age, gender, etc. and using Poisson re-
gression framework, determined which of these factors are signiﬁcant to the claims frequency.
The identiﬁed factors are then used in the a priori ratemaking of bonus-malus scales based
on the Belgian bonus-malus system. In the system, the number of annual claims are as-
sumed to conform to a Poisson distribution. They then suggested that proposing separate
bonus-malus scales with diﬀerent premiums based on the diﬀerent a priori characteristics
will help in ensuring that a a policyholder obtains fair premium in the system with respect
to his speciﬁc characteristic risk.
1.5.4 “Bonus-Malus Systems: “Lack of Transparency” and Adequacy Measure”
(2002) by Paola Verico
In this particular article, Verico introduces a formula to calculate the “adequacy” of a bonus-
malus system, which he deﬁned as the system’s “capability to bring every policyholder up
to the point where he pays a premium fair to him”. The formula is given by the square of
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the mean of the error the BMS makes, which is equal to:
∞

0

s

i=1
a
i
(λ)
c
i
λ
0
C(∞)
−λ

2
u(λ)dλ (2)
where λ
0
is the portfolio mean claim frequency, a
i
(λ) is the probability of a policyholder
with claim frequency λ to belong to class i, u(λ) is the true density of the portfolio,
c
i
C(∞)
is
the premium due of the policyholder in class i at the steady state of the system, in the long
run. Values of c
i
that would minimize this error are to be derived to achieve the adequacy
of the system.
1.6 Methodology
1.6.1 Data gathering
Data from a microﬁnance company X such as its loan repayment schemes and contact in-
formation of its borrowers will be obtained. A suﬃciently large sample of borrower will be
surveyed. Data such as age, gender, business-type, frequency of late repayments, etc. will
be gathered from the survey.
1.6.2 Modelling the system
A speciﬁc loan repayment scheme of the microﬁnance company X will be modeled using
the methods presented by Aug´e et al (2010) to determine the eﬃciency of the scheme. If
a problem of overindebtedness is identiﬁed, regression analysis of the data gathered will be
used to determine the signiﬁcant factors that aﬀect the loan repayment problems. A model
similar to that constructed by Mokhtar et al. (2012) will be used.
1.6.3 Formulation of the bonus-malus scales
The a priori rate-making scheme of Pitrebois et al. (2003) will be used. The bonus-malus
systems for the speciﬁc loan repayment scheme analyzed will be characterized by the follow-
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ing:
1. The classes will be deﬁned by the amount of regular dividend payments to be paid
during the term of the loan.
2. The transition determinants between classes will be deﬁned by the frequency of late
repayments during the previous loan.
Diﬀerent premiums will be computed for diﬀerent bonus-malus scales in terms of the diﬀer-
ent characteristics of the borrowers which were proved previously to aﬀect loan repayment
problems.
1.6.4 Development of new loan policies
New loan policies consisting of bonus-malus payments will be developed based on the formu-
lated bonus-malus scales and the calculated premiums. The adequacy of the bonus-malus
systems will be measured using the formula proposed by Verico (2002).
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2 Plan of Work
2.1 Schedule of Activities
Phase Description of activity Duration in weeks Expected output
Data gathering Finding a microﬁnance in-
stitution X, producing ques-
tionnaires and surveying a
sample of borrowers
10 Data on the loan re-
payment schemes of X
and the characteristics
of its borrowers
Modelling the system Formulation and revision of
parameters and estimators
to analyze the system
3 The factors aﬀecting
loan repayment prob-
lems of the system will
be determined.
Construction and de-
velopment of bonus-
malus scales to min-
imize loan repayment
problems
Overall formulation of and
revision of models needed.
8 Final ouput
Production of ﬁnal pa-
per
Includes checking with ad-
viser and revisions
3 Final paper for publi-
cation
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2.2 Financial Plan
Detailed Line-Item Budget
Particular Amount in pesos Amount in pesos
Sem 1 Sem 2
Travel 1000.00 0.00
Supplies 1000.00 1000.00
Sundries/Other Services 5000.00 1000.00
Subtotals 7000.00 2000.00
Total Amount (in pesos): 9000.00
References
[1] L´eo Aug´e, Aurore Lebrun, and Ana¨ıs Piozin, Microcredit models and Yunus equation.
2010.
[2] Linda Mayoux,Micro-ﬁnance and the empowerment of women : a review of the key issues.
2000.
[3] Suraya Hanim Mokhtar, Gilbert Nartea, and Christopher Gan, Determinants of micro-
credit loans repayment problem among microﬁnance borrowers in Malaysia. 2012
[4] Sandra Pitrebois, Michel Denuit, and Jean-Francois Walhin, Marketing and Bonus Malus
Systems. 2003.
[5] Paola Verico, Bonus-Malus Systems: “Lack of Transparency” and Adequacy Mea-
sure.2002.
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